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Properties

Label 2-720-80.29-c1-0-17
Degree $2$
Conductor $720$
Sign $0.986 + 0.161i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.39 − 0.258i)2-s + (1.86 + 0.719i)4-s + (−2.19 + 0.404i)5-s − 1.81·7-s + (−2.40 − 1.48i)8-s + (3.16 + 0.00742i)10-s + (0.331 − 0.331i)11-s + (0.0310 − 0.0310i)13-s + (2.52 + 0.469i)14-s + (2.96 + 2.68i)16-s − 1.00i·17-s + (−2.08 − 2.08i)19-s + (−4.39 − 0.828i)20-s + (−0.547 + 0.375i)22-s + 6.22·23-s + ⋯
L(s)  = 1  + (−0.983 − 0.183i)2-s + (0.933 + 0.359i)4-s + (−0.983 + 0.180i)5-s − 0.686·7-s + (−0.851 − 0.524i)8-s + (0.999 + 0.00234i)10-s + (0.100 − 0.100i)11-s + (0.00860 − 0.00860i)13-s + (0.674 + 0.125i)14-s + (0.740 + 0.671i)16-s − 0.243i·17-s + (−0.477 − 0.477i)19-s + (−0.982 − 0.185i)20-s + (−0.116 + 0.0800i)22-s + 1.29·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.986 + 0.161i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.986 + 0.161i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.694438 - 0.0563147i\)
\(L(\frac12)\) \(\approx\) \(0.694438 - 0.0563147i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.258i)T \)
3 \( 1 \)
5 \( 1 + (2.19 - 0.404i)T \)
good7 \( 1 + 1.81T + 7T^{2} \)
11 \( 1 + (-0.331 + 0.331i)T - 11iT^{2} \)
13 \( 1 + (-0.0310 + 0.0310i)T - 13iT^{2} \)
17 \( 1 + 1.00iT - 17T^{2} \)
19 \( 1 + (2.08 + 2.08i)T + 19iT^{2} \)
23 \( 1 - 6.22T + 23T^{2} \)
29 \( 1 + (-6.28 - 6.28i)T + 29iT^{2} \)
31 \( 1 - 7.11T + 31T^{2} \)
37 \( 1 + (-0.0723 - 0.0723i)T + 37iT^{2} \)
41 \( 1 + 3.06iT - 41T^{2} \)
43 \( 1 + (3.78 + 3.78i)T + 43iT^{2} \)
47 \( 1 - 10.0iT - 47T^{2} \)
53 \( 1 + (-7.04 - 7.04i)T + 53iT^{2} \)
59 \( 1 + (-6.68 + 6.68i)T - 59iT^{2} \)
61 \( 1 + (-2.89 - 2.89i)T + 61iT^{2} \)
67 \( 1 + (-0.150 + 0.150i)T - 67iT^{2} \)
71 \( 1 + 14.5iT - 71T^{2} \)
73 \( 1 - 15.0T + 73T^{2} \)
79 \( 1 - 15.5T + 79T^{2} \)
83 \( 1 + (-5.48 + 5.48i)T - 83iT^{2} \)
89 \( 1 + 14.3iT - 89T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.51164962663077856058607137613, −9.418346365740569242198027005527, −8.719515003776894978204027878069, −7.928876283554670476378899795114, −6.92795660953598780310035709602, −6.47756164305081780411119901362, −4.84749361645760021437676340009, −3.49915819164909565185687769465, −2.70721539944571431913894124635, −0.78481839423319970229399819681, 0.816156749821622410473461126377, 2.61526100523465630492781657737, 3.77123793156016913322447323819, 5.08240145627882333675848181185, 6.43886293840811718672341983214, 6.93729239079875551788548985609, 8.137885714667196585900240670786, 8.479685682366884088796551272757, 9.611652622972356001252202043038, 10.24518091043525899498265085325

Graph of the $Z$-function along the critical line